Abstract

Suppose are positive integers. Let be the space of all complex upper triangular matrices, and let be an injective linear map on . Then is an idempotent matrix in whenever is an idempotent matrix in if and only if there exists an invertible matrix such that , or when , , where or and or

1. Introduction

Suppose are positive integers. Let be the space of all complex matrices, and let be all upper triangular in . For , , we denote by their tensor product (a.k.a. Kronecker product).

Linear preserver problem is a hot area in matrix and operator theory; there are many results about this area (see [1–14]). Specially, the idempotence preservers and the rank one preservers play an important role (see [1, 2]); therefore, it is meaningful to study the idempotence preservers. Chan et al. [3] first characterize linear transformations on preserving idempotent matrices. Šemrl [4] applying projective geometry gives the form of transformations on rank-1 idempotents. Tang et al. [5] investigate injective linear idempotence preservers on .

In quantum information science, quantum states of a system with physical states are represented as density matrices, that is, positive semidefinite matrices with trace one. If and are two quantum states in two quantum systems, then describes a joint state in bipartite system . Recently, many researchers consider the problem combining linear preserver problem with quantum information science. They determine the structure of linear maps on by using information only about the images of matrices possessing tensor product form. One can see [15–18] and their references for some background on linear preserver problems on tensor spaces arising in quantum information science.

Inspired by the above, the purpose of this paper is to study injective linear maps on satisfying is an idempotent matrix whenever is an idempotent matrix in . If we remove the assumption that map is injective, then may have various forms as follows.

Example 1. is a linear idempotent preserver on .

Example 2. is a linear idempotent preserver on .

We end this section by introducing some notations which will be used in the following sections. Let be the complex field, the identity matrix, the zero matrix whose order is omitted in different matrices just for simplicity, and (resp., ) the transpose (resp., rank) of . , stands for the matrix with at the th entry and otherwise. Denote by (also ) the matrix . Clearly, if , then . For positive integers and with , let be the set of all integers between and . For any , we define by
For any , we define and such that (it is easy to see that and are well defined). It is easy to see that
We define a partial ordering of by if and only if and . We say that and are comparable, if or .

2. Preliminary Results

We need the form of injective linear idempotence preserver on , which was obtained in [5].

Lemma 3 (see [5, Theorem 1]). Let be an injective linear map on . Then is an idempotent matrix in whenever is an idempotent matrix in if and only if there exists an invertible matrix such that
where or .

It is clear that . For example, and
It is easy to see that and . In fact, we can point out the positions of elements which are in .

The next Lemma describes the partial ordering we defined in Section 1, which is useful to prove our main Theorem.

Lemma 5 (see [19, Theorem 1]). Let , and let be a matrix with rows and columns containing all elements of . If every two elements of in the same row and column are comparable, respectively, then there exist permutation matrices and such that
or when or
or when
or when

The following lemmas would make the proof of the main theorem more concise.

Lemma 6. Suppose is an idempotent matrix such that also is an idempotent matrix. Then there exists an idempotent matrix such that .

Proof. By and , we have
Set
where . Then (6) implies
Hence, ; therefore, the lemma holds.

Lemma 7. Let be an idempotent matrix such that also is an idempotent matrix. Then there exists an idempotent such that .

Proof. It follows from , is idempotent that
Let
where , , then (10) implies
Hence, , , , . Similarly, from being idempotent, we have .

Lemma 9 (see [6, Page 62, Exercise 1]). Suppose are idempotent matrices such that, for any , is idempotent. Let . Then there exists an invertible matrix such that
where is the diagonal matrix in which all diagonal entries are zero except those in the st to the th rows.

Lemma 10. Let be idempotent matrices of rank-1 such that for any , is idempotent. Then there exist a permutation on and an invertible matrix such that

Proof. By being an idempotent matrix of rank-1, we can assume
where with zero diagonal entries. It follows from , being is idempotent that , . Hence, is a permutation on . By , we can see . Let ; then and
By being idempotent, we obtain . Let ; then
Continuing to do this, we can find . Let ; then
This completes the proof.

3. The Main Result

The main result of this paper is as follows.

Theorem 11. Suppose are positive integers and is an injective linear map on . Then is an idempotent matrix in whenever is an idempotent matrix in if and only if there exists an invertible matrix such that
or when
where, for , or .

Proof. The sufficiency is obvious. We will prove the necessity by the following six steps.Step 1. There exist a permutation on and an invertible matrix such that
Proof of Step 1. By Lemma 10, we only need to prove that
And for any ,
It follows from and are idempotent matrices in that and , are idempotent matrices. We obtain by using Lemma 9 that there exists an invertible matrix such that
where and . For any , it follows from and being idempotent matrices in that and are idempotent matrices; we obtain by (22) and Lemma 6 that
where is an idempotent matrix. For any , is an idempotent matrix in ; we have is an idempotent matrix. It follows from (23) that is an idempotent matrix. If , by Lemma 9, we can obtain that there exists some such that . This, together with (23), implies , which is a contradiction to the fact that is injective. Hence, , . By Lemma 9, there exists an invertible matrix such that . Let ; it follows from (23) that
Hence, (20) and (21) hold. This completes the proof of Step 1.By Step 1, we may assume that for any , ,
From this, together with (3), we can also write
Step 2. (i) For any , , and are comparable.(ii) For any , , and are comparable.Proof of Step 2. (i) Suppose there exist some and such that and are not comparable. Without loss of generality, we may assume that
It follows that
For any , by and being idempotent matrices in , we obtain by (25) that
are idempotent matrices in ; hence, by Lemma 8
From (27), by Lemma 4, we obtain that , which is a contradiction to the fact that is injective. Using a similar method, we may prove (ii) holds. This completes the proof of Step 2.Note. It is easy to see that is a bijective map from to with from to . This, together with , is a permutation on ; we obtain that
Let ; then forms an matrix containing all elements of . Step 2 implies that every two elements of in the same row and column are comparable, respectively. Thus, applying Lemma 5 to , we conclude that one of – holds. If holds, then, for any but fixed , all , , are in the same row; that is, , and ; hence, it follows from (26) that
Similarly, if holds, then and, for any but fixed , all , , are in the same row; that is, , and ; hence, it follows from (26) that
We claim that and do not hold. Indeed, if holds, for convenience, we assume and we first consider the special case , in (one can use a similar method to prove the case of ). Thus, by (26), we have
Since, for any , , and , are idempotent matrices in , we obtain by (34) that
are idempotent matrices in . This, together with Lemma 8, implies
For any , is an idempotent matrix in . Thus, by (34) and (36)
Similarly, since for any , , and , are idempotent matrices in , we have
For any , is an idempotent matrix in ; we obtain
It follows from (37) and (39) that , which is a contradiction to that is injective.For general case, by , we can choose a permutation of and a permutation of such that
From this, together with (26), we obtain
Using a similar method as the above, we can drive a contradiction. Similarly, we may prove does not hold.If (32) holds, we may assume
where is a permutation on . If (33) holds, we may assume
We next assume (42) to prove of theorem holds and one can use similar methods to prove of theorem if (43) holds.Step 3. There exists an invertible matrix such that
where, for , or .Proof of Step 3. For any idempotent matrix , since and are idempotent matrices in , we obtain by (42) that
are idempotent matrices in . By Lemma 7, we have
where is an idempotent matrix. By the arbitrariness of , we can expand to be a linear map on . Hence
It is easy to see that is injective and preserving idempotents. Thus, by Lemma 3, there exists an invertible such that , where or . Let ; we complete the proof of this step.By Step 3, we may assume
Step 4. or .Proof of Step 4. If , then this claim is clear. For , we prove that if , then or . Otherwise, we assume (other cases can be proven by using similar methods). Since for any , and are idempotent matrices in , we have by using (48) that and are idempotent matrices in . This, together with Lemma 8, implies
Similarly,
By being an idempotent matrix in , we obtain by using (48), (49), and (50) that
is an idempotent matrix in ; that is,
This implies , which is a contradiction. Hence, we complete the proof of Step 4.Step 5. For any , ,and there exists such that
Proof of Step 5. We prove the case of (one can use a similar method to prove the case of ). Hence,
Without loss of generality, we may assume , . Since for any , and are idempotent matrices in , we obtain by (54) that and are idempotent matrices in . This, together with Lemma 8, implies , where . Let ; then
Let ; by (55), one can obtain
Let , . Then, (56) turn into
By (57), using a similar method to Step 3, one can obtain
where or . Hence
Thus,
This implies
From , , one can easily see that and ; thus . This completes the proof of Step 5.By Step 5, we may assume .Step 6. For and , we have . Proof of Step 6. From is an idempotent matrix in , we have
is an idempotent matrix in . It follows from or that .When , let ; then
Hence,
When , let ; then
Thus
This completes the proof of the theorem.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the referee for his/her careful reading of the paper and valuable comments which greatly improved the readability of the paper. Li Yang is supported by Vocational Education Institute in Heilongjiang Province “12th five-year development plan” and the Guiding Function and Practice Research of Mathematical Modeling in Advanced Mathematics Teaching of New Rise Financial Institutions (Grant no. GG0666). Jinli Xu was supported by the National Natural Science Foundation Grants of China (Grant no. 11171294) and the Natural Science Foundation of Heilongjiang Province of China (Grant no. A201013).